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Abstract

One width and overlay system (which can be employed with that of U. S. Patent 3,957,376) does not provide consistent results in measuring the width of lines with gradients using algorithms involving the Fourier transform of the diffraction pattern of the line.

Country

United States

Language

English (United States)

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Algorithm for Line Width Extraction from Cosine Transform

One width and overlay system (which can be employed with that of U. S.
Patent 3,957,376) does not provide consistent results in measuring the width of
lines with gradients using algorithms involving the Fourier transform of the
diffraction pattern of the line.

One conventional algorithm, the "Burke" algorithm, smooths and
differentiates one side of the diffraction pattern, takes the modulus of the Fourier
transform of the smoothed, differentiated pattern, and assigns the component
with the highest amplitude in the modulus to the line width.

Analysis of the system has shown that the Fourier transform has terms
representing b, a-b, a, and a + b (Fig. 1). The coefficients of these terms are
functions of the amplitude and phase of the complex reflectances of the different
parts of the pattern on the wafers, and it is not generally true that the line-width
term has the greatest amplitude, as previously assumed. In fact, for example,
over the range of nominal tolerances in the film thicknesses (of photoresist and
the underlying SiO(2) film), the coefficient of the line-width term, a, is predicted to
be zero over a wide range. In particular, this coefficient is predicted to be zero at
several points within the operating range of tolerances. In practice, the Burke
algorithm has been observed to give unreliable results, and it has displayed the
predictable behavior of switching between peaks in the transform, as the
amplitudes of the peaks change as a result of film thickness variations. The
peaks that are chosen in error are usually those representing the line-width-plus-
gradient (a+b) or the line-width-minus-gradient (a-b) peaks. Consequently, the
Burke algorithm is characterized as having an uncertainty approximately equal to
the gradient width.

A more rigorous analysis of the Fourier transform is diagrammed in Fig. 2.
The diffraction data is multiplied by a quadratic function of the diode number, the
DC component is removed, and the cosine transform of the diffraction pattern is
found. On the assumption that the diffraction pattern is symmetrical, the sine
transform is expected to be zero and is discarded so the cosine transform is the
whole transform. Fig. 3 is a representation of a cosine transform amplitude
versus line-width spectrum. The component b (gradient width) has an amplitude
-Rho cos Phi, the component (a-b), line-width-minus gradient, has an amplitude
of -1/2, the component a, line-width, has an amplitude +Rho cos Phi, and the
component (a+b), line-width plus gradient, has an amplitude -Rho/2//2. The
value Rho is the ratio of reflectances inside and outside the line, and Phi is the
phase angle between reflectances inside and outside the line. The values of Rho
and Phi vary significantly over the expected film thickness tolerance variations.
Consequently, the amplitude of the line-width component varies between + Rho
and - Rho.